In Mysterious Pattern, Math and Nature Converge (2013) • Hacker News Board

Re: In Mysterious Pattern, Math and Nature Converge (2013)

#11

Post by rndn » Thu, Feb 26, 2015, 5:43 PM UTC

squidfood wrote: post #7

rndn wrote: post #5

Earlier quoted context omitted.

Here are the slides of a talk on universality that Terry Tao gave some time ago: https://terrytao.files.wordpress.com/2011/01/universality.pd... It's basically the observation that large systems converge to exhibit rather simple behaviors. It generalizes other reoccurring patterns such as power laws, Euler's number, the normal distribution and the fibonacci sequence. For some reason nature is surprisingly frugal rega…

Well, sure. I say "oh look, when animals breed, it has this curve" and then I say "oh look, when money is invested it grows with this curve." The insight is that "multiplication works". This helps, because I don't need to re-invent logarithms and e to study animal population growth. And of course, different rules lead to different math (be it power laws, the Normal distribution, universiality, Phi, whatever). This is…

I'm not too familiar with all of this (perhaps I'm even misinterpreting it), but I think they do realize that it would be problematic to throw population trajectories and interest rates into the same pot. I think the motivation is rather to find reasons why things do not behave completely unpredictable at larger scales and I cannot think of good reasons against efforts to find more and more general descriptions of that.

Re: In Mysterious Pattern, Math and Nature Converge (2013)

#12

Post by squidfood » Thu, Feb 26, 2015, 6:13 PM UTC

rndn wrote: post #11

squidfood wrote: post #7

Earlier quoted context omitted.

Well, sure. I say "oh look, when animals breed, it has this curve" and then I say "oh look, when money is invested it grows with this curve." The insight is that "multiplication works". This helps, because I don't need to re-invent logarithms and e to study animal population growth. And of course, different rules lead to different math (be it power laws, the Normal distribution, universiality, Phi, whatever). This is…

I'm not too familiar with all of this (perhaps I'm even misinterpreting it), but I think they do realize that it would be problematic to throw population trajectories and interest rates into the same pot. I think the motivation is rather to find reasons why things do not behave completely unpredictable at larger scales and I cannot think of good reasons against efforts to find more and more general descriptions of th…

I agree with the motivation and interest! The issue comes when people try too hard to find patterns to fit a preconceived hypothesis.

I maintain a large (publicly available) ecological dataset, and my data have been drawn into several meta-analyses of this type. Often the idea is to simply see if my data empirically fit the "right" distribution. And they fit it and say "wow, it's all connected".

But then I look at the Bus data in this example (the +'s that represent actual data). I'm guessing I could fit a lognormal distribution, a Gamma distribution, or the Dyson distribution that they actuall use, and the data wouldn't be enough to distinguish between them.

Now, all of these distributions result from "simple" rules, but they are three very different sets of simple rules. For the Bus data, the "repelling" by little slips of paper makes sense as the mechanism, so it's a good hypothesis.

But then to flip that around, and say "since this distribution fits the ecological data, the underlying mechanism must be Bus Repelling" is wholly unjustified (as there are other possible fits). And there's a lot of junk science that does that.

Re: In Mysterious Pattern, Math and Nature Converge (2013)

#13

Post by westoncb » Fri, Feb 27, 2015, 4:54 PM UTC

drostie wrote: post #10

westoncb wrote: post #8

Interesting! Reading a bit more on the Wikipedia article, my understanding is that 'Universality' is an appropriate name because it describes classes of extremely diverse systems that can be described by the same abstract model—which also always happens to be a scale-invarient model that resembles a physical phase transition. Is that right, anybody? I thought this list of systems with the same 'universality class' wa…

Close but not quite. The term "universality" is different in phase transition theory from what it means in random matrix theory (which is what's at play here), but they've got some similarities too. In phase transition theories, you've got two different states (like liquid water and water vapor), and when you vary some high-level parameters (like pressure and temperature) you can go from one of these states to the ot…

Hey, thanks a lot! Didn't expect such a great reply. I was still a bit fuzzy on a couple points, if you don't mind. I'm visiting another country and haven't found reliable wifi yet, hence slow reply time.

I think I get the scale invariant theory concept from a mathematical perspective, but I don't see why this would be the case: "Because you've got this considerable mixing of the two states, often "zooming in" is the same as, say, adjusting the proportion of liquid to gas"

Regarding random matrices, my understanding now is that given a random matrix of sufficient size, under a suitable definition of "random," if we look at the eigenvalue density function it's always going to be (roughly?) the same--we at least know it will be semi-circular. Further, there exist physical systems that can be modeled by random matrices; and there's a mapping between eigenvalues of the matrix and certain physical characteristics of the system. So, knowing the density function is always the same for these random matrices, we can assume certain shared characteristics of any systems that can be modeled by a random matrix.

In random matrices and phase transitions we would like to know how certain macroscopic parameters will behave, given some data like a matrix, or state of a phase transition. But, in both cases our starting data contains a lot of essentially irrelevant data that these theories prescribe a method for filtering out, since it assures us that knowledge of the symmetries involved are all that will matter.

Am I close? :)